@HJRW: factor means quotient.

You may find the proof of the local finiteness of groups of exponent four in Derek J. S. Robinson's group theory book published by Springer.

so I would like to know how you arrived to the point that in the generic case The stabilizer is trivial?

As far as I know for the orbit of a $k$-subset $S$ of $\mathbb{Z}_p$, $\lambda=\frac{k(k-1)}{|Stab_{AL(p)}(S)|}$, where $Stab_{AL(p)}(S)=\{\alpha\in AL(p) \;|\; S^\alpha=S\}$. So you are claiming that $|Stab_{AL(p)}(S)|=1$. Could you please give a hint for this?

This may be interesting to know that the converse is true; that is, $d(G)\leq max_p(d(S_p))+1$ for any finite group $G$, where $S_p$ denotes a Sylow $p$-subgroup of $G$. See [R. M. Guralnick, On the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), 521–523] and [A. Lucchini, A bound on the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), 313–317.]

@ArturoMagidin: Regular 2-groups are abelian. So your second example is not regular.